We show that every compactum has cohomological dimension $1$ with respect
to a finitely generated nilpotent group $G$ whenever it has cohomological
dimension $1$ with respect to the abelianization of $G$. This is applied
to the extension theory to obtain a cohomological dimension theory condition
for a finite-dimensional compactum $X$ for extendability of every map from
a closed subset of $X$ into a nilpotent $\CW$-complex $M$ with finitely
generated homotopy groups over all of $X$.